October  2017, 22(8): 3199-3220. doi: 10.3934/dcdsb.2017170

Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems

1. 

Faculty of Mathematics and Computer Science, West University of Timişoara, Pârvan Blvd. No. 4,300223 Timişoara, Romania

2. 

Academy of Romanian Scientists, Splaiul Independenţei 54,050094, Bucharest, Romania

Received  February 2016 Revised  October 2016 Published  June 2017

The aim of this paper is to present a new and very general method for the study of the uniform exponential trichotomy of nonautonomous dynamical systems defined on the whole axis. We consider a discrete dynamical system and we introduce the property of $(r, p)$-admissibility relative to an associated control system, where $r, p∈ [1, ∞]$. In several constructive steps, we obtain full descriptions of the sufficient conditions and respectively of the necessary criteria for uniform exponential trichotomy based on the $(r, p)$-admissibility of the pair $(\ell^∞(\mathbb{Z}, X), \ell^1(\mathbb{Z}, X))$. In the same time, we provide a complete diagram of the $\ell^p$-spaces which can be considered in the admissible pairs for the study of the uniform exponential trichotomy of discrete dynamical systems. We present illustrative examples in order to motivate the hypotheses and the generality of our method. Finally, we apply the main results to obtain new criteria for uniform exponential trichotomy of dynamical systems modeled by evolution families using the admissibility of various pairs of $\ell^p$-spaces.

Citation: Adina Luminiţa Sasu, Bogdan Sasu. Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3199-3220. doi: 10.3934/dcdsb.2017170
References:
[1]

C. V. Coffman and J. J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166.  doi: 10.1007/BF01350095.

[2]

S. Elaydi and O. Hájek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362-374.  doi: 10.1016/0022-247X(88)90255-7.

[3]

S. Elaydi and O. Hájek, Exponential dichotomy and trichotomy of nonlinear differential equations, Differ. Integral Equ., 3 (1990), 1201-1224. 

[4]

S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Differ. Equations Appl., 3 (1998), 417-448.  doi: 10.1080/10236199708808113.

[5]

J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, 1966.

[6]

N. Van Minh, On the proof of characterisations of the exponential dichotomy, Proc. Amer. Math. Soc., 127 (1999), 779-782.  doi: 10.1090/S0002-9939-99-04640-7.

[7]

N. Van Minh and N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44.  doi: 10.1006/jmaa.2001.7450.

[8]

N. Van Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations, 198 (2004), 381-421.  doi: 10.1016/j.jde.2003.10.006.

[9]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.  doi: 10.1016/0022-0396(84)90082-2.

[10]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.  doi: 10.1090/S0002-9939-1988-0958058-1.

[11]

K. J. Palmer, Shadowing and Silnikov chaos, Nonlinear Anal., 27 (1996), 1075-1093.  doi: 10.1016/0362-546X(95)00042-T.

[12]

O. Perron, Die Stabilitätsfrage bei Differentialgleischungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.

[13]

V. A. Pliss and G. R. Sell, Robustness of the exponential dichotomy in infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 3 (1999), 471-513.  doi: 10.1023/A:1021913903923.

[14]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems Lecture Notes in Mathematics, vol. 2002, Springer, 2010. doi: 10.1007/978-3-642-14258-1.

[15]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Operator Theory, 73 (2012), 107-151.  doi: 10.1007/s00020-012-1959-7.

[16]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423.

[17]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems, III, J. Differential Equations, 22 (1976), 497-522.  doi: 10.1016/0022-0396(76)90043-7.

[18]

B. Sasu and A. L. Sasu, Exponential dichotomy and $(\ell^p, \ell^q) $-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408.  doi: 10.1016/j.jmaa.2005.04.047.

[19]

B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478.  doi: 10.1016/j.jmaa.2005.12.002.

[20]

B. Sasu and A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z., 253 (2006), 515-536.  doi: 10.1007/s00209-005-0920-8.

[21]

A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920.  doi: 10.1016/j.jmaa.2008.03.019.

[22]

A. L. Sasu and B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17-32.  doi: 10.1016/j.jmaa.2011.02.045.

[23]

B. Sasu and A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057-3084.  doi: 10.3934/dcds.2013.33.3057.

[24]

A. L. Sasu and B. Sasu, Discrete admissibility and exponential trichotomy of dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 2929-2962.  doi: 10.3934/dcds.2014.34.2929.

[25]

A. L. Sasu and B. Sasu, Admissibility and exponential trichotomy of dynamical systems described by skew-product flows, J. Differential Equations, 260 (2016), 1656-1689.  doi: 10.1016/j.jde.2015.09.042.

show all references

References:
[1]

C. V. Coffman and J. J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166.  doi: 10.1007/BF01350095.

[2]

S. Elaydi and O. Hájek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362-374.  doi: 10.1016/0022-247X(88)90255-7.

[3]

S. Elaydi and O. Hájek, Exponential dichotomy and trichotomy of nonlinear differential equations, Differ. Integral Equ., 3 (1990), 1201-1224. 

[4]

S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Differ. Equations Appl., 3 (1998), 417-448.  doi: 10.1080/10236199708808113.

[5]

J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, 1966.

[6]

N. Van Minh, On the proof of characterisations of the exponential dichotomy, Proc. Amer. Math. Soc., 127 (1999), 779-782.  doi: 10.1090/S0002-9939-99-04640-7.

[7]

N. Van Minh and N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44.  doi: 10.1006/jmaa.2001.7450.

[8]

N. Van Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations, 198 (2004), 381-421.  doi: 10.1016/j.jde.2003.10.006.

[9]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.  doi: 10.1016/0022-0396(84)90082-2.

[10]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.  doi: 10.1090/S0002-9939-1988-0958058-1.

[11]

K. J. Palmer, Shadowing and Silnikov chaos, Nonlinear Anal., 27 (1996), 1075-1093.  doi: 10.1016/0362-546X(95)00042-T.

[12]

O. Perron, Die Stabilitätsfrage bei Differentialgleischungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.

[13]

V. A. Pliss and G. R. Sell, Robustness of the exponential dichotomy in infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 3 (1999), 471-513.  doi: 10.1023/A:1021913903923.

[14]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems Lecture Notes in Mathematics, vol. 2002, Springer, 2010. doi: 10.1007/978-3-642-14258-1.

[15]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Operator Theory, 73 (2012), 107-151.  doi: 10.1007/s00020-012-1959-7.

[16]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423.

[17]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems, III, J. Differential Equations, 22 (1976), 497-522.  doi: 10.1016/0022-0396(76)90043-7.

[18]

B. Sasu and A. L. Sasu, Exponential dichotomy and $(\ell^p, \ell^q) $-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408.  doi: 10.1016/j.jmaa.2005.04.047.

[19]

B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478.  doi: 10.1016/j.jmaa.2005.12.002.

[20]

B. Sasu and A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z., 253 (2006), 515-536.  doi: 10.1007/s00209-005-0920-8.

[21]

A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920.  doi: 10.1016/j.jmaa.2008.03.019.

[22]

A. L. Sasu and B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17-32.  doi: 10.1016/j.jmaa.2011.02.045.

[23]

B. Sasu and A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057-3084.  doi: 10.3934/dcds.2013.33.3057.

[24]

A. L. Sasu and B. Sasu, Discrete admissibility and exponential trichotomy of dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 2929-2962.  doi: 10.3934/dcds.2014.34.2929.

[25]

A. L. Sasu and B. Sasu, Admissibility and exponential trichotomy of dynamical systems described by skew-product flows, J. Differential Equations, 260 (2016), 1656-1689.  doi: 10.1016/j.jde.2015.09.042.

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